Contacts in totally separable packings in the plane and in high dimensions

Naszódi, Márton and Swanepoel, Konrad J. ORCID: 0000-0002-1668-887X (2022) Contacts in totally separable packings in the plane and in high dimensions. Journal of Computational Geometry, 13 (1). pp. 471-483. ISSN 1920-180X

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Abstract

We study the contact structure of totally separable packings of translates of a convex body K in Rd, that is, packings where any two translates of the packing have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number Hsep(K) of K is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of K. We show that for each d ≥ 8, there exists a smooth and strictly convex K in Rd with Hsep(K) > 2d, and asymptotically, Hsep(K) = Ω((3/√8)d). We show that Alon’s packing of Euclidean unit balls such that each translate touches at least 2√d others whenever d is a power of 4, can be adapted to give a totally separable packing of translates of the ℓ1-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of n translates of any planar convex body K. We prove that the maximum equals ⌊2n − 2√n⌋ if and only if K is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.

Item Type:	Article
Additional Information:	© 2022 The Author(s).
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited:	23 Nov 2022 17:30
Last Modified:	08 Apr 2024 04:48
URI:	http://eprints.lse.ac.uk/id/eprint/117416

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